Quim. Nova, Vol. 41, No. 5, 587-593, 2018 http://dx.doi.org/10.21577/0100-4042.20170198

EXPLAINING THE GEOMETRY OF SIMPLE MOLECULES USING MOLECULAR ORBITAL ENERGY-LEVEL DIAGRAMS BUILT BY USING SYMMETRY PRINCIPLES

Sérgio P. Machado and Roberto B. Faria* Departamento de Química Inorgânica, Instituto de Química, Universidade Federal do Rio de Janeiro, 21941-909 Rio de Janeiro – RJ, Brasil

Recebido em 27/08/2017; aceito em 08/01/2018; publicado na web em 08/02/2018 Educação

The built of qualitative energy-level molecular diagrams for different geometries of simple molecules allow to explain the preferred geometry. The diagrams are built using simple symmetry principles and explain, on basis of the number of nonbonding electrons, for example, why the molecule of water is bent and not linear and ammonia is pyramidal and not planar. This simple energy principle does not need to consider the Valence Shell Electron-Pair Repulsion theory (VSEPR theory) neither hybrid orbitals to explain the geometry of simple molecules. This discussion is more appropriate to inorganic chemistry courses where symmetry is a common topic.

Keywords: symmetry; group theory; molecular orbitals; molecular geometry

INTRODUCTION Other works deal with the same subject we present in this 15 article. Mulliken presents a discussion on NH3 which considers this

The geometries of molecules like H2O, NH3 and CH4 are usually molecule in the geometries pyramidal and plane trigonal. However, explained, at undergraduate level, in general and inorganic chemistry he concludes that the pyramidal geometry is the preferred because courses, using the Valence Shell Electron-Pair Repulsion theory of the mixing between s and pz nitrogen in bonding molecular (VSEPR theory) and hybrid orbitals.1-6 In the case of the VSEPR orbitals of the pyramidal geometry. Baird16 presents a discussion the starting point is the Lewis structure of each compound. It is on the geometries of several AXn molecules which is based on considered that electrons groups around the central atom will be as Walsh diagrams for the highest occupied molecular orbitals. far as possible, producing the observed molecular geometry (linear for Miller and Ellison17 also present Walsh diagrams calculated by a two electrons groups, triangular for three electrons groups, tetrahedral computational chemistry software for a series of AXn molecules, for four electrons groups, and so on). In the case of the molecules allowing discussing their geometries. indicated above, all of them have four electron groups but some of In this work we present the use of qualitative molecular orbital these have lone pairs resulting in the H2O bent, NH3 pyramidal, and energy-level diagrams built from simple group theory principles

CH4 tetrahedral geometry, as it is well known. of symmetry. However, differently from the articles cited above, When using hybrid orbitals, the central atom in all these molecules our approach is based on the different occupancy of nonbonding is considered to use sp3 orbitals, which corresponds to the tetrahedral molecular orbitals for each geometry of the molecule considered. This geometry. In this case some sp3 hybrid orbitals contain lone pair of allows explain the preferred geometry of simple molecules without electrons, resulting in the observed geometry for the cited molecules. requiring the building of complete Walsh diagram. Our approach is On the other hand, in the inorganic chemistry courses, when simpler and is based mainly in the number of electrons on nonbonding part of the time is dedicated to the application of symmetry to build molecular orbitals. the symmetry-adapted linear combinations of atomic orbitals, the Symmetry is a common subject in inorganic chemistry stability of these molecules can be discussed more properly based courses (most commonly third or fourth semester or even later) on the molecular orbital energy-level diagrams for these molecules. and undergraduate students are very familiar with it. In this way, Several inorganic chemistry textbooks present a very detailed we have used the approach described below in the undergraduate molecular orbital energy-level diagrams for several simple molecules, inorganic chemistry course. We observed that the comparison of for example, BeH2, CO2, and H2O, justifying their geometries based the energy-level molecular diagrams for different geometries of an on the occupancy of the molecular orbitals. However, as far we specific molecule is a very significant application of symmetry and know, these textbooks do not present a discussion of what happen molecular orbital theory which gives a very strong support to more with the molecular energy if we consider a molecule with a different deep discussions of chemical bond based on molecular orbitals. The geometry.4-6 More specialized books on spectroscopy,7 present a inclusion of this subject allowed show for the students a simple discussion based on the change of the energy of molecular orbitals as and direct application of symmetry and group theory on the very the geometry of a molecule is gradually changed, for example, in the basic topic of molecular geometry together with the presentation case of AH2 species, changing from linear to bent geometry. This kind of fundamental aspects of symmetry. This was a very significant of discussion is based on qualitative Walsh diagrams.8-14 These Walsh improvement because we start to show an application of these diagrams describe how the energy of each molecular orbital changes principles earlier than before. Students, usually, are very anxious as the molecular geometry is modified between two possibilities as, about where they will apply this knowledge, which is abstract, for example, linear and bent geometries. These gradual modifications mathematical and spatial reasoning, being difficult for most part on the energy of the molecular orbitals, however, in some cases, are of the class. Before we have included this subject, the students not easy to forecast, making difficult the discussion in class. need wait for the first application of symmetry until the discussing the electronic spectroscopy of transition metal complexes. As a *e-mail: [email protected] consequence, we observed a very clear improvement in the number 588 Machado and Faria Quim. Nova of the students which obtained higher grades in the questions related with the basic principles of symmetry and group theory. We did not apply a tool to measure this improvement but we estimated that 70 to 80% of the class have increased their comprehension level. The starting point of this approach is the identification of the irreducible representation of the atomic orbitals. After that, the energy level diagram of the molecular orbitals is built, combining the atomic orbitals of the same irreducible representation. Then, the molecular orbitals are filled with the electrons using the aufbau principle. Comparison of the number of electrons which occupy bonding, antibonding, and nonbonding orbitals, at each different geometry of the same molecule, allows explain its preferred molecular geometry. In the following we consider that the reader knows the basic principles of Group Theory applied to chemistry including symmetry operations and the use of the Character Tables.4-6,18-22

THE WATER MOLECULE

This molecule has only two possible geometries indicated in Figure 2. Results for the application of the Ĉ2 operator (on the z axis), of

Figure 1: linear (D∞h) and bent (C2v). The convention for the coordinate the (C2v) point group, over the 2px, 2py, and 2pz oxygen orbitals of the water system and axis is the same used by Orchin and Jaffé,7 in which the molecule x axis is perpendicular to the plane of the molecule.

Figure 1. The two possible geometries for the water molecule: linear (D∞h) and bent (C2v)

In the case of the bent geometry the molecule belongs to the point group C2v and its Character Table allows to identify the irreducible representation of each oxygen (the central atom) atomic orbital based Figure 3. Results for the application of the operator , of the (C2v) point on the last column indications (see Table 1). The 2s orbital has a group, over the 2px, 2py, and 2pz oxygen orbitals of the water molecule spherical symmetry and belongs to the totally symmetric irreducible representation which is always the first to appear in any Character

Table, in this case A1. The orbitals 2px, 2py, and 2pz belongs to B1,

B2, and A1, respectively. How these assignments are made, is shown below for the symmetry operations of the group C2v, as an example. For other point groups this can be found in many textbooks which show how to find the irreducible representation of the atomic orbitals in a molecule.4,5,8-11

In the case of the Ĉ2, , and symmetry operators the orbitals 2px, 2py, and 2pz transform as shown in Figures 2, 3, and 4, respectively. To identify the irreducible representation for each orbital the result of the application of the symmetry operator is taken as equal to 1 if the orbital does not change and the result is -1 if the orbital changes its signal. In this way, Table 1 shows the results and the attribution of an irreducible representation for each oxygen atomic orbital 2px, 2py, and 2pz. For the 1s atomic orbitals on each hydrogen (peripheral atoms), indicated by sH1 and sH2, they cannot be considered separately. As the

Ĉ2 symmetry operator exchange both 1s atomic orbitals, they must Figure 4. Results for the application of the operator , of the (C2v) point be considered together using a two-dimensional matrix. In the case group, over the 2px, 2py, and 2pz oxygen orbitals of the water molecule Vol. 41, No. 5 Explaining the geometry of simple molecules 589

Table 1. Applying the C2v symmetry operations to oxygen 2px, 2py, and 2pz atomic orbitals in the water molecule

(xz) (yz) C2v E C2 σv σ′v

A1 1 1 1 1 z

A2 1 1 −1 −1

B1 1 −1 1 −1 x

B2 1 −1 −1 1 y

2px 1 −1 1 −1 B1

2py 1 −1 −1 1 B2

2pz 1 1 1 1 A1 of the identity symmetry operation, the matrix representation of this operation can be indicated as

This notation can be simplified considering that the first column Figure 5. Qualitative molecular orbital energy-level diagram for water in and first line of the matrices refer to the sH1 atomic orbital and the the bent geometry (C2v) second column and second line refer to the sH2 atomic orbital. In addition, the character or the trace of the matrix (χ, the sum of the Now, let us consider the possibility that the water molecule is main diagonal elements) is also indicated. linear. In this case the molecule will belongs to the point group D∞h (Figure 1). Using the information on the last column of the Character Table (Table 3) we can assign the oxygen atomic orbitals to the + following irreducible representations: 2s (Σg , the totally symmetric + representation), 2pz (Σu ), and 2px and 2py together (Πu).

In the case of the hydrogen 1s atomic orbitals, any Ĉ2 perpendicular to the principal axis (z) will exchange the hydrogens 1s orbitals in the atoms H1 and H2 and because of this they must be worked together using two dimensional matrixes, as shown below.

These results show that 1s atomic orbitals of the hydrogens produce a reducible representation which can be shown to be formed by the sum A1 + B2 (see Table 2).

Table 2. Applying the C2v symmetry operations to hydrogen 1s atomic orbitals in the water molecule

(xz) (yz) C2v E C2 σv σ′v

A1 1 1 1 1 z

A2 1 1 −1 −1

B1 1 −1 1 −1 x

B2 1 −1 −1 1 y

H1 H2 (s ,s ) 2 0 0 2 A1 + B2 These results show that 1s hydrogen orbitals (sH1,sH2) produce a reducible representation which can be shown that is formed by + + With this information we can set up the energy level diagram Σg + Σu (see Table 3). for the molecular orbitals of water (Figure 5), considering that only Using these results we can build the molecular orbital energy- atomic orbitals of the same symmetry species can combine. The level diagram for the water molecule in the linear geometry (D∞h), as symmetry labels are written in lower case when indicating atomic is shown in Figure 6. Note that now the 2px and 2py oxygen atomic and molecular orbitals. Note also that the 2px oxygen atomic orbital orbitals do not have any hydrogen orbitals with the same irreducible does not have any hydrogen orbitals to combine and because of this representation to combine and because of this they stay as a πu it stays as a b1 nonbonding molecular orbital which has the same nonbonding molecular orbital which has the same energy level of energy level of the 2px atomic orbital. the 2px and 2py atomic orbitals. 590 Machado and Faria Quim. Nova

Table 3. Applying the D∞h symmetry operations to the hydrogen 1s atomic orbitals of the linear water molecule

ϕ ϕ D∞h E 2C∞ … ∞σv i 2S∞ … ∞C2

+ Σg 1 1 … 1 1 1 … 1

- Σg 1 1 … −1 1 1 … −1

Πg 2 2cosϕ … 0 2 −2cosϕ … 0

∆g 2 2cos2ϕ … 0 2 2cos2ϕ … 0

+ Σu 1 1 … 1 −1 −1 … −1 z

- Σu 1 1 … −1 −1 −1 … 1

Πu 2 2cosϕ … 0 −2 2cosϕ … 0 (x,y)

∆u 2 2cos2ϕ … 0 −2 −2cos2ϕ … 0

H1 H2 + + (s ,s ) 2 2 2 0 0 0 Σg + Σu

presented in Figure 5 than the presence of two sp3 nonbonding equivalent orbitals.23

THE AMMONIA MOLECULE

The same approach used with the water molecule can be applied to explain the most stable geometry of other simple molecules. Let

us now consider the NH3 molecule in two different geometries:

pyramidal (C3v) and planar (D3h) (Figure 7). Application of the same symmetry principles produces the results shown in Table 4.

Figure 6. Qualitative molecular orbital energy-level diagram for water in Figure 7. The two possible geometries for the ammonia molecule: pyramidal the linear geometry (D∞h) (C3v) and planar (D3h)

Comparison between Figures 5 and 6 shows that in the linear Table 4. Applying the C3v symmetry operations to the atomic orbitals of the NH molecule geometry the water molecule has four electrons in two πu nonbonding 3 molecular orbitals, but in the bent geometry it has only two electrons C3v E 2C3 3σv in the b1 nonbonding molecular orbital, being more stable in this geometry because it has more electrons in lower energy levels. A1 1 1 1 z

Clearly the reason for the bent geometry of the water molecule is A2 1 1 −1 that one of the πu nonbonding molecular orbitals in the linear geometry E 2 −1 0 (x,y) becomes a bonding molecular orbital in the bent geometry, decreasing H1 H2 H3 (s ,s ,s ) 3 0 1 A1 + E the energy of the molecule. The other πu nonbonding molecular orbital stays as a b1 nonbonding. 8,16,17 Some authors discuss in more detail the transformation of From these results we can say that both nitrogen 2s and 2pz one set of orbitals in the other, as the geometry is changed from linear atomic orbitals belong to the irreducible representation A1 and the to bent, using the very well-known Walsh diagrams. This is a more orbitals 2px and 2py belongs to the irreducible representation E. detailed discussion which is necessary to explain the geometry of The three hydrogen 1s orbitals produce a reducible representation, species with a lower number of electrons, like BeH2, CH2, NH2, and which decomposes to A1 + E. With this information the qualitative + BH2 , which are linear. In the case of water, it is necessary consider molecular orbital energy-level diagram for NH3 can be assembled as only the number of nonbonding electrons in both geometry. shown in Figure 8, as it is presented in many textbooks, without any It is worth to say that the conclusion that water is bent was nonbonding molecular orbital. obtained without consider the repulsion between the electrons pairs in Let us now consider the NH3 in the planar geometry (D3h). the VSEPR theory or by the use of sp3 hybrid orbitals. In addition, the Application of the principles of symmetry produces the results presence of three low energy peaks in the photoelectron spectrum of shown in Table 5, which indicates that the nitrogen 2s atomic orbital water is in a better agreement with the molecular orbital energy-level belongs to the irreducible representation A1′ (totally symmetric), Vol. 41, No. 5 Explaining the geometry of simple molecules 591

Figure 8. Qualitative molecular orbital energy-level diagram for NH3 in the Figure 9. Qualitative molecular orbital energy-level diagram for NH3 in the pyramidal geometry (C3v) planar geometry (D3h)

Table 5. Applying the D3h symmetry operations to the atomic orbitals of the planar NH3 molecule

D3h E 2C3 3C2 σh 2S3 3σv

A1′ 1 1 1 1 1 1

A2′ 1 1 –1 1 1 –1 E′ 2 –1 0 2 –1 0 (x,y)

A1″ 1 1 1 –1 –1 –1

A2″ 1 1 –1 –1 –1 1 z E″ 2 –1 0 –2 1 0 (sH1,sH2,sH3) 3 0 1 3 0 1 A + E 1′ ′ Figure 10. The two possible geometries for the methane molecule: square

planar (D4h) and tetrahedral (Td) the 2pz belongs to A2″, and the orbitals 2px and 2py belongs to a two- dimensional irreducible representation E′. Table 6. Applying the Td symmetry operations to the atomic orbitals of the The three hydrogen 1s orbitals produce a reducible representation tetrahedral CH4 molecule which is formed by A1′ + E′. Using this information the molecular Td E 8C3 3C2 6S4 6σd orbital energy-level diagram for NH3 in the planar geometry (D3h) can be built, as shown in Figure 9, which presents one electron pair A1 1 1 1 1 1 in the a2″ nonbonding molecular orbital. A2 1 1 1 –1 –1 Comparison between Figures 8 and 9 shows that NH3 in the E 2 –1 2 0 0 planar geometry has two electrons in the a2″ nonbonding molecular T 3 0 –1 1 –1 orbitals, but in the pyramidal geometry it does not have electrons 1 T 3 0 –1 –1 1 (x,y,z) in nonbonding orbitals, being more stable in this geometry because 2 (sH1,sH2,sH3,sH4) 4 1 0 0 2 A + T it has more electrons in lower energy levels. Again the reason for 1 2 the pyramidal geometry of the NH3 molecule is the lower energy of the valence electrons in this geometry compared with the planar As can be seen from Figures 11 and 12 the methane in the geometry, independent from the lower repulsion between the planar geometry (D4h) has one pair of electrons in the nonbonding 3 electrons pairs in the VSEPR theory or by the use of sp hybrid a2u molecular orbital, but in the tetrahedral geometry there is no orbitals. nonbonding molecular orbital occupied, which gives a lower energy

for the Td geometry than D4h geometry. THE METHANE MOLECULE BENT AND LINEAR THREE ATOM SPECIES WITH We can now consider the tetrahedral and planar geometry of DOUBLE BONDS methane, which belongs to the point groups Td and D4h, respectively (Figure 10). Tables 6 and 7 and Figures 11 and 12 present the results In addition to these simple cases above, it is intriguing, for for the application of symmetry concepts and the molecular orbital example, why is the ozone molecule bent and CO2 linear? Table 8 energy-level diagrams for both cases. shows several simple three atoms species and their electron valence 592 Machado and Faria Quim. Nova

Table 7. Applying the D4h symmetry operations to the atomic orbitals of the planar CH4 molecule

D4h E 2C4 C2 2C2′ 2C2″ i 2S4 σh 2σv 2σd

A1g 1 1 1 1 1 1 1 1 1 1

A2g 1 1 1 –1 –1 1 1 1 –1 –1

B1g 1 –1 1 1 –1 1 –1 1 1 –1

B2g 1 –1 1 –1 1 1 –1 1 –1 1

Eg 2 0 –2 0 0 2 0 –2 0 0

A1u 1 1 1 1 1 –1 –1 –1 –1 –1

A2u 1 1 1 –1 –1 –1 –1 –1 1 1 z

B1u 1 –1 1 1 –1 –1 1 –1 –1 1

B2u 1 –1 1 –1 1 –1 1 –1 1 –1

Eu 2 0 –2 0 0 –2 0 2 0 0 (x,y) H1 H2 H3 H4 (s ,s ,s ,s ) 4 0 0 2 0 0 0 4 2 0 A1g + B1g + Eu

Figure 11. Qualitative molecular orbital energy-level diagram for CH4 in the Figure 12. Qualitative molecular orbital energy-level diagram for CH4 in tetrahedral geometry (Td) the planar geometry (D4h) counting. As can be seen, only sixteen electron species can be linear 9 (D∞h) as it was observed by other authors. To explain the geometries of the species shown in Table 8 using molecular orbital energy-level diagrams the situation now is more complex because instead of the terminal hydrogen atoms we have now oxygen or nitrogen atoms. It means that in addition to the s atomic orbitals, the terminal atoms have three p orbitals, which makes more difficult to produce simple molecular orbital energy-level diagrams. However, we can make a simple analysis considering only the π frontier molecular, as it is shown in Figure 13 for the CO2 molecule.

Table 8. Trinuclear AX2 species and their electron valence counting and geometry

Number of valence linear bent electrons - 16 CO2, N3 • 17 NO2 - 18 O3, NO2 , SO2 • 19 ClO2 Figure 13. Simplified molecular orbital energy-level diagram for CO2 (16 - 20 ClO2 valence electrons) showing only the frontier π orbitals Vol. 41, No. 5 Explaining the geometry of simple molecules 593

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ACKNOWLEDGEMENT

The authors thank the financial support by Conselho Nacional de Desenvolvimento Científico e Tecnológico-CNPq (Grants no. 306.050/2016-1 and 304.423/2014-9).

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